Integrand size = 21, antiderivative size = 119 \[ \int (a+a \sin (c+d x))^2 \tan ^5(c+d x) \, dx=-\frac {31 a^2 \log (1-\sin (c+d x))}{8 d}-\frac {a^2 \log (1+\sin (c+d x))}{8 d}-\frac {2 a^2 \sin (c+d x)}{d}-\frac {a^2 \sin ^2(c+d x)}{2 d}+\frac {a^4}{4 d (a-a \sin (c+d x))^2}-\frac {9 a^3}{4 d (a-a \sin (c+d x))} \] Output:
-31/8*a^2*ln(1-sin(d*x+c))/d-1/8*a^2*ln(1+sin(d*x+c))/d-2*a^2*sin(d*x+c)/d -1/2*a^2*sin(d*x+c)^2/d+1/4*a^4/d/(a-a*sin(d*x+c))^2-9/4*a^3/d/(a-a*sin(d* x+c))
Time = 0.30 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.63 \[ \int (a+a \sin (c+d x))^2 \tan ^5(c+d x) \, dx=-\frac {a^2 \left (31 \log (1-\sin (c+d x))+\log (1+\sin (c+d x))-\frac {2}{(-1+\sin (c+d x))^2}-\frac {18}{-1+\sin (c+d x)}+16 \sin (c+d x)+4 \sin ^2(c+d x)\right )}{8 d} \] Input:
Integrate[(a + a*Sin[c + d*x])^2*Tan[c + d*x]^5,x]
Output:
-1/8*(a^2*(31*Log[1 - Sin[c + d*x]] + Log[1 + Sin[c + d*x]] - 2/(-1 + Sin[ c + d*x])^2 - 18/(-1 + Sin[c + d*x]) + 16*Sin[c + d*x] + 4*Sin[c + d*x]^2) )/d
Time = 0.28 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.91, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3042, 3186, 99, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \tan ^5(c+d x) (a \sin (c+d x)+a)^2 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \tan (c+d x)^5 (a \sin (c+d x)+a)^2dx\) |
| \(\Big \downarrow \) 3186 |
| \(\displaystyle \frac {\int \frac {a^5 \sin ^5(c+d x)}{(a-a \sin (c+d x))^3 (\sin (c+d x) a+a)}d(a \sin (c+d x))}{d}\) |
| \(\Big \downarrow \) 99 |
| \(\displaystyle \frac {\int \left (\frac {a^4}{2 (a-a \sin (c+d x))^3}-\frac {9 a^3}{4 (a-a \sin (c+d x))^2}+\frac {31 a^2}{8 (a-a \sin (c+d x))}-\frac {a^2}{8 (\sin (c+d x) a+a)}-\sin (c+d x) a-2 a\right )d(a \sin (c+d x))}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {a^4}{4 (a-a \sin (c+d x))^2}-\frac {9 a^3}{4 (a-a \sin (c+d x))}-\frac {1}{2} a^2 \sin ^2(c+d x)-2 a^2 \sin (c+d x)-\frac {31}{8} a^2 \log (a-a \sin (c+d x))-\frac {1}{8} a^2 \log (a \sin (c+d x)+a)}{d}\) |
Input:
Int[(a + a*Sin[c + d*x])^2*Tan[c + d*x]^5,x]
Output:
((-31*a^2*Log[a - a*Sin[c + d*x]])/8 - (a^2*Log[a + a*Sin[c + d*x]])/8 - 2 *a^2*Sin[c + d*x] - (a^2*Sin[c + d*x]^2)/2 + a^4/(4*(a - a*Sin[c + d*x])^2 ) - (9*a^3)/(4*(a - a*Sin[c + d*x])))/d
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] && (IntegerQ[p] | | (GtQ[m, 0] && GeQ[n, -1]))
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*tan[(e_.) + (f_.)*(x_)]^(p _.), x_Symbol] :> Simp[1/f Subst[Int[x^p*((a + x)^(m - (p + 1)/2)/(a - x) ^((p + 1)/2)), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && E qQ[a^2 - b^2, 0] && IntegerQ[(p + 1)/2]
Result contains complex when optimal does not.
Time = 3.56 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.64
| method | result | size |
| risch | \(4 i a^{2} x +\frac {a^{2} {\mathrm e}^{2 i \left (d x +c \right )}}{8 d}+\frac {i a^{2} {\mathrm e}^{i \left (d x +c \right )}}{d}-\frac {i a^{2} {\mathrm e}^{-i \left (d x +c \right )}}{d}+\frac {a^{2} {\mathrm e}^{-2 i \left (d x +c \right )}}{8 d}+\frac {8 i a^{2} c}{d}+\frac {i \left (-9 a^{2} {\mathrm e}^{i \left (d x +c \right )}-16 i a^{2} {\mathrm e}^{2 i \left (d x +c \right )}+9 a^{2} {\mathrm e}^{3 i \left (d x +c \right )}\right )}{2 \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )^{4} d}-\frac {a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{4 d}-\frac {31 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{4 d}\) | \(195\) |
| derivativedivides | \(\frac {a^{2} \left (\frac {\sin \left (d x +c \right )^{8}}{4 \cos \left (d x +c \right )^{4}}-\frac {\sin \left (d x +c \right )^{8}}{2 \cos \left (d x +c \right )^{2}}-\frac {\sin \left (d x +c \right )^{6}}{2}-\frac {3 \sin \left (d x +c \right )^{4}}{4}-\frac {3 \sin \left (d x +c \right )^{2}}{2}-3 \ln \left (\cos \left (d x +c \right )\right )\right )+2 a^{2} \left (\frac {\sin \left (d x +c \right )^{7}}{4 \cos \left (d x +c \right )^{4}}-\frac {3 \sin \left (d x +c \right )^{7}}{8 \cos \left (d x +c \right )^{2}}-\frac {3 \sin \left (d x +c \right )^{5}}{8}-\frac {5 \sin \left (d x +c \right )^{3}}{8}-\frac {15 \sin \left (d x +c \right )}{8}+\frac {15 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+a^{2} \left (\frac {\tan \left (d x +c \right )^{4}}{4}-\frac {\tan \left (d x +c \right )^{2}}{2}-\ln \left (\cos \left (d x +c \right )\right )\right )}{d}\) | \(206\) |
| default | \(\frac {a^{2} \left (\frac {\sin \left (d x +c \right )^{8}}{4 \cos \left (d x +c \right )^{4}}-\frac {\sin \left (d x +c \right )^{8}}{2 \cos \left (d x +c \right )^{2}}-\frac {\sin \left (d x +c \right )^{6}}{2}-\frac {3 \sin \left (d x +c \right )^{4}}{4}-\frac {3 \sin \left (d x +c \right )^{2}}{2}-3 \ln \left (\cos \left (d x +c \right )\right )\right )+2 a^{2} \left (\frac {\sin \left (d x +c \right )^{7}}{4 \cos \left (d x +c \right )^{4}}-\frac {3 \sin \left (d x +c \right )^{7}}{8 \cos \left (d x +c \right )^{2}}-\frac {3 \sin \left (d x +c \right )^{5}}{8}-\frac {5 \sin \left (d x +c \right )^{3}}{8}-\frac {15 \sin \left (d x +c \right )}{8}+\frac {15 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+a^{2} \left (\frac {\tan \left (d x +c \right )^{4}}{4}-\frac {\tan \left (d x +c \right )^{2}}{2}-\ln \left (\cos \left (d x +c \right )\right )\right )}{d}\) | \(206\) |
| parts | \(\frac {a^{2} \left (\frac {\tan \left (d x +c \right )^{4}}{4}-\frac {\tan \left (d x +c \right )^{2}}{2}+\frac {\ln \left (1+\tan \left (d x +c \right )^{2}\right )}{2}\right )}{d}+\frac {a^{2} \left (\frac {\sin \left (d x +c \right )^{8}}{4 \cos \left (d x +c \right )^{4}}-\frac {\sin \left (d x +c \right )^{8}}{2 \cos \left (d x +c \right )^{2}}-\frac {\sin \left (d x +c \right )^{6}}{2}-\frac {3 \sin \left (d x +c \right )^{4}}{4}-\frac {3 \sin \left (d x +c \right )^{2}}{2}-3 \ln \left (\cos \left (d x +c \right )\right )\right )}{d}+\frac {2 a^{2} \left (\frac {\sin \left (d x +c \right )^{7}}{4 \cos \left (d x +c \right )^{4}}-\frac {3 \sin \left (d x +c \right )^{7}}{8 \cos \left (d x +c \right )^{2}}-\frac {3 \sin \left (d x +c \right )^{5}}{8}-\frac {5 \sin \left (d x +c \right )^{3}}{8}-\frac {15 \sin \left (d x +c \right )}{8}+\frac {15 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )}{d}\) | \(215\) |
Input:
int((a+a*sin(d*x+c))^2*tan(d*x+c)^5,x,method=_RETURNVERBOSE)
Output:
4*I*a^2*x+1/8*a^2/d*exp(2*I*(d*x+c))+I*a^2/d*exp(I*(d*x+c))-I*a^2/d*exp(-I *(d*x+c))+1/8*a^2/d*exp(-2*I*(d*x+c))+8*I/d*a^2*c+1/2*I*(-9*a^2*exp(I*(d*x +c))-16*I*a^2*exp(2*I*(d*x+c))+9*a^2*exp(3*I*(d*x+c)))/(exp(I*(d*x+c))-I)^ 4/d-1/4*a^2/d*ln(exp(I*(d*x+c))+I)-31/4*a^2/d*ln(exp(I*(d*x+c))-I)
Time = 0.11 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.41 \[ \int (a+a \sin (c+d x))^2 \tan ^5(c+d x) \, dx=\frac {4 \, a^{2} \cos \left (d x + c\right )^{4} + 22 \, a^{2} \cos \left (d x + c\right )^{2} - 12 \, a^{2} - {\left (a^{2} \cos \left (d x + c\right )^{2} + 2 \, a^{2} \sin \left (d x + c\right ) - 2 \, a^{2}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 31 \, {\left (a^{2} \cos \left (d x + c\right )^{2} + 2 \, a^{2} \sin \left (d x + c\right ) - 2 \, a^{2}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \, {\left (4 \, a^{2} \cos \left (d x + c\right )^{2} - 5 \, a^{2}\right )} \sin \left (d x + c\right )}{8 \, {\left (d \cos \left (d x + c\right )^{2} + 2 \, d \sin \left (d x + c\right ) - 2 \, d\right )}} \] Input:
integrate((a+a*sin(d*x+c))^2*tan(d*x+c)^5,x, algorithm="fricas")
Output:
1/8*(4*a^2*cos(d*x + c)^4 + 22*a^2*cos(d*x + c)^2 - 12*a^2 - (a^2*cos(d*x + c)^2 + 2*a^2*sin(d*x + c) - 2*a^2)*log(sin(d*x + c) + 1) - 31*(a^2*cos(d *x + c)^2 + 2*a^2*sin(d*x + c) - 2*a^2)*log(-sin(d*x + c) + 1) - 2*(4*a^2* cos(d*x + c)^2 - 5*a^2)*sin(d*x + c))/(d*cos(d*x + c)^2 + 2*d*sin(d*x + c) - 2*d)
\[ \int (a+a \sin (c+d x))^2 \tan ^5(c+d x) \, dx=a^{2} \left (\int 2 \sin {\left (c + d x \right )} \tan ^{5}{\left (c + d x \right )}\, dx + \int \sin ^{2}{\left (c + d x \right )} \tan ^{5}{\left (c + d x \right )}\, dx + \int \tan ^{5}{\left (c + d x \right )}\, dx\right ) \] Input:
integrate((a+a*sin(d*x+c))**2*tan(d*x+c)**5,x)
Output:
a**2*(Integral(2*sin(c + d*x)*tan(c + d*x)**5, x) + Integral(sin(c + d*x)* *2*tan(c + d*x)**5, x) + Integral(tan(c + d*x)**5, x))
Time = 0.03 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.81 \[ \int (a+a \sin (c+d x))^2 \tan ^5(c+d x) \, dx=-\frac {4 \, a^{2} \sin \left (d x + c\right )^{2} + a^{2} \log \left (\sin \left (d x + c\right ) + 1\right ) + 31 \, a^{2} \log \left (\sin \left (d x + c\right ) - 1\right ) + 16 \, a^{2} \sin \left (d x + c\right ) - \frac {2 \, {\left (9 \, a^{2} \sin \left (d x + c\right ) - 8 \, a^{2}\right )}}{\sin \left (d x + c\right )^{2} - 2 \, \sin \left (d x + c\right ) + 1}}{8 \, d} \] Input:
integrate((a+a*sin(d*x+c))^2*tan(d*x+c)^5,x, algorithm="maxima")
Output:
-1/8*(4*a^2*sin(d*x + c)^2 + a^2*log(sin(d*x + c) + 1) + 31*a^2*log(sin(d* x + c) - 1) + 16*a^2*sin(d*x + c) - 2*(9*a^2*sin(d*x + c) - 8*a^2)/(sin(d* x + c)^2 - 2*sin(d*x + c) + 1))/d
Time = 0.23 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.71 \[ \int (a+a \sin (c+d x))^2 \tan ^5(c+d x) \, dx=-\frac {1}{8} \, a^{2} {\left (\frac {\log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{d} + \frac {31 \, \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{d} + \frac {4 \, {\left (d \sin \left (d x + c\right )^{2} + 4 \, d \sin \left (d x + c\right )\right )}}{d^{2}} - \frac {2 \, {\left (9 \, \sin \left (d x + c\right ) - 8\right )}}{d {\left (\sin \left (d x + c\right ) - 1\right )}^{2}}\right )} \] Input:
integrate((a+a*sin(d*x+c))^2*tan(d*x+c)^5,x, algorithm="giac")
Output:
-1/8*a^2*(log(abs(sin(d*x + c) + 1))/d + 31*log(abs(sin(d*x + c) - 1))/d + 4*(d*sin(d*x + c)^2 + 4*d*sin(d*x + c))/d^2 - 2*(9*sin(d*x + c) - 8)/(d*( sin(d*x + c) - 1)^2))
Time = 17.35 (sec) , antiderivative size = 283, normalized size of antiderivative = 2.38 \[ \int (a+a \sin (c+d x))^2 \tan ^5(c+d x) \, dx=\frac {4\,a^2\,\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}{d}-\frac {a^2\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )}{4\,d}-\frac {\frac {15\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{2}-22\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+\frac {61\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{2}-36\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+\frac {61\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{2}-22\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+\frac {15\,a^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-12\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+14\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-12\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )}-\frac {31\,a^2\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-1\right )}{4\,d} \] Input:
int(tan(c + d*x)^5*(a + a*sin(c + d*x))^2,x)
Output:
(4*a^2*log(tan(c/2 + (d*x)/2)^2 + 1))/d - (a^2*log(tan(c/2 + (d*x)/2) + 1) )/(4*d) - ((61*a^2*tan(c/2 + (d*x)/2)^3)/2 - 22*a^2*tan(c/2 + (d*x)/2)^2 - 36*a^2*tan(c/2 + (d*x)/2)^4 + (61*a^2*tan(c/2 + (d*x)/2)^5)/2 - 22*a^2*ta n(c/2 + (d*x)/2)^6 + (15*a^2*tan(c/2 + (d*x)/2)^7)/2 + (15*a^2*tan(c/2 + ( d*x)/2))/2)/(d*(8*tan(c/2 + (d*x)/2)^2 - 4*tan(c/2 + (d*x)/2) - 12*tan(c/2 + (d*x)/2)^3 + 14*tan(c/2 + (d*x)/2)^4 - 12*tan(c/2 + (d*x)/2)^5 + 8*tan( c/2 + (d*x)/2)^6 - 4*tan(c/2 + (d*x)/2)^7 + tan(c/2 + (d*x)/2)^8 + 1)) - ( 31*a^2*log(tan(c/2 + (d*x)/2) - 1))/(4*d)
Time = 0.19 (sec) , antiderivative size = 413, normalized size of antiderivative = 3.47 \[ \int (a+a \sin (c+d x))^2 \tan ^5(c+d x) \, dx=\frac {a^{2} \left (2 \,\mathrm {log}\left (\tan \left (d x +c \right )^{2}+1\right ) \sin \left (d x +c \right )^{4}-4 \,\mathrm {log}\left (\tan \left (d x +c \right )^{2}+1\right ) \sin \left (d x +c \right )^{2}+2 \,\mathrm {log}\left (\tan \left (d x +c \right )^{2}+1\right )+12 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1\right ) \sin \left (d x +c \right )^{4}-24 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1\right ) \sin \left (d x +c \right )^{2}+12 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1\right )-27 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \sin \left (d x +c \right )^{4}+54 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \sin \left (d x +c \right )^{2}-27 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+3 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) \sin \left (d x +c \right )^{4}-6 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) \sin \left (d x +c \right )^{2}+3 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )-2 \sin \left (d x +c \right )^{6}-8 \sin \left (d x +c \right )^{5}+\sin \left (d x +c \right )^{4} \tan \left (d x +c \right )^{4}-2 \sin \left (d x +c \right )^{4} \tan \left (d x +c \right )^{2}+9 \sin \left (d x +c \right )^{4}+25 \sin \left (d x +c \right )^{3}-2 \sin \left (d x +c \right )^{2} \tan \left (d x +c \right )^{4}+4 \sin \left (d x +c \right )^{2} \tan \left (d x +c \right )^{2}-6 \sin \left (d x +c \right )^{2}-15 \sin \left (d x +c \right )+\tan \left (d x +c \right )^{4}-2 \tan \left (d x +c \right )^{2}\right )}{4 d \left (\sin \left (d x +c \right )^{4}-2 \sin \left (d x +c \right )^{2}+1\right )} \] Input:
int((a+a*sin(d*x+c))^2*tan(d*x+c)^5,x)
Output:
(a**2*(2*log(tan(c + d*x)**2 + 1)*sin(c + d*x)**4 - 4*log(tan(c + d*x)**2 + 1)*sin(c + d*x)**2 + 2*log(tan(c + d*x)**2 + 1) + 12*log(tan((c + d*x)/2 )**2 + 1)*sin(c + d*x)**4 - 24*log(tan((c + d*x)/2)**2 + 1)*sin(c + d*x)** 2 + 12*log(tan((c + d*x)/2)**2 + 1) - 27*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**4 + 54*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**2 - 27*log(tan((c + d*x)/2) - 1) + 3*log(tan((c + d*x)/2) + 1)*sin(c + d*x)**4 - 6*log(tan((c + d*x)/2) + 1)*sin(c + d*x)**2 + 3*log(tan((c + d*x)/2) + 1) - 2*sin(c + d *x)**6 - 8*sin(c + d*x)**5 + sin(c + d*x)**4*tan(c + d*x)**4 - 2*sin(c + d *x)**4*tan(c + d*x)**2 + 9*sin(c + d*x)**4 + 25*sin(c + d*x)**3 - 2*sin(c + d*x)**2*tan(c + d*x)**4 + 4*sin(c + d*x)**2*tan(c + d*x)**2 - 6*sin(c + d*x)**2 - 15*sin(c + d*x) + tan(c + d*x)**4 - 2*tan(c + d*x)**2))/(4*d*(si n(c + d*x)**4 - 2*sin(c + d*x)**2 + 1))